Why do Italian road signs use the infinitive tense in their warning, while, for example, those in English use the imperative?
Turn off lights
Spegnere le luci (and not "Spegnete le luci" or "Spegni le luci")
Or, maybe, is there a rule that says that the infinitive tense can be work as an imperative? If so, can anyone suggest some cases in which that rule is used in everyday speech?
The infinitive mood is commonly used for expressing rules especially in signs (of any kind, not just road signs).
For instance
Non fumare
Non calpestare il prato
Tenere la destra
The language "trick" behind this use of the infinitive form is the omission of the clause Si prega di or equivalent, so the above sentences are read as
Si prega di non fumare
Si prega di non calpestare il prato
Si prega di tenere la destra
Such form is not used in everyday's spoken language, as it's a convention used for giving orders and stating rules in an impersonal and formal way.
That being said, there's an official use of the infinitive mood as imperative, which is the negative imperative.
In Italian the positive imperative form goes as follows
Tieni la destra!
Parla con lei!
whereas the negative imperative is formed with non + infinitive mood, as in
Non tenere la destra!
Non parlare con lei!
As discussed in the comments, it's also nice to notice the differences and the similarities with other Romance languages, such as Spanish and French.
Apparently French has the same identical construct as Italian for expressing formal impersonal orders, for instance
Ne pas fumer
Non fumare
which is again a shortening for
Merci de ne pas fumer
Grazie di non fumare or more idiomatically Si prega di non fumare
On the other hand Spanish behaves differently and it doesn't have a special construct for impersonal orders, rather just using the formal imperative form, which is formed with the subjunctive
No fume
Non fumare, but also Non fumi
or
Reduzca la velocidad
Ridurre la velocità, but also Riduca la velocità
Another reason is related to the fact that the Italian imperative must be conjugated. For example,
Turn off the lights
might be translated, using the imperative, as:
Spegni would sound rude, because of the use of the informal 2nd singular person, and it wouldn't be an appropriate way to address someone having a higher status than the author of the command, or even a stranger (regardless of his/her social status).
Spegnete would sound awkward if there were a single person in the room (e.g., in a bathroom).
Spenga might seem appropriate, but it would actually be a somewhat weird way to address, e.g., a child (children are not usually addressed by using the formal third-person singular).
As a result, the most appropriate tense to be used in road signs is the infinitive, which doesn't refer to a particular number or social status of the addressed people. Note that, as far as I know, the use of the infinitive tense to give orders derives from Latin, because the second-person singular of the Latin imperative, in the passive form, is the same as the present infinitive.
As others already said, you could understand the sentence as starting with si prega di, but truly the infinitive is used to express commands in Italian. That is true for phrases like ritirare lo scontrino alla cassa or non ridere, which uses the negative form.
It's probably a shortening of "Si prega di spegnere le luci", which is more polite than the imperative form.
I think a big reason could be that the infinitive conveys a greater sense of authority. For example a football trainer might urge the players to run by shouting "Correte!" (imperative) or make it sound more of an order by saying "Correre!" (infinitive). It might be the same reason why in the military it's more common to issue orders using the infinitive.
